5. RDF-Compatible Model-Theoretic Semantics (Normative)

This model-theoretic semantics for OWL
is an extension of the semantics defined in
the RDF semantics
[RDF Semantics], and
defines the OWL
semantic
extension of RDF.

NOTE: There is a strong
correspondence between the semantics for OWL DL defined in this
section and the Direct Model-Theoretic Semantics defined in Section 3 (see Theorem 1 and Theorem 2 in Section 5.4). If, however, any conflict
should ever arise between these two forms, then the Direct
Model-Theoretic Semantics takes precedence.

5.1. The OWL and RDF universes

All of the OWL vocabulary is defined on the 'OWL universe', which is a
division of part of the RDF universe into three parts,
namely OWL individuals, classes, and properties.
The class extension of owl:Thing
comprises the individuals of the OWL universe.
The class extension of owl:Class comprises the
classes of the OWL universe.
The union of the class extensions of
owl:ObjectProperty,
owl:DatatypeProperty,
owl:AnnotationProperty, and
owl:OntologyProperty
comprises the properties of the OWL universe.

There are two different styles of using OWL.
In the more free-wheeling style, called OWL Full,
the three parts of the OWL universe
are identified with their RDF counterparts,
namely the class extensions of
rdfs:Resource, rdfs:Class, and
rdf:Property.
In OWL Full, as in RDF, elements of the OWL universe can be both an
individual and a class, or, in fact, even an individual, a class, and a
property.
In the more restrictive style, called OWL DL here,
the three parts are different from their RDF
counterparts and, moreover, pairwise disjoint.
The more-restrictive OWL DL style gives up some expressive power in return for
decidability of entailment.
Both styles of OWL provide entailments that are missing in a naive
translation of the DAML+OIL model-theoretic semantics into the RDF
semantics.

A major difference in practice between the two styles lies in the care
that is required to ensure that URI references are actually in the
appropriate part of the OWL universe.
In OWL Full, no care is needed.
In OWL DL, localizing information must be provided for many of the URI
references used.
These localizing assumptions are all trivially true in OWL Full,
and can also be ignored when one uses the OWL abstract
syntax, which corresponds closely to OWL DL.
But when writing OWL DL in triples, however, close attention must be paid
to which elements of the vocabulary belong to which part of the OWL
universe.

5.2. OWL Interpretations

The semantics of OWL DL and
OWL Full are very similar.
The common portion of their semantics is thus given first, and the
differences left until later.

From the RDF semantics
[RDF Semantics],
for V a set of URI references and literals
containing the RDF and RDFS vocabulary and D a
datatype map,
a D-interpretation of V is a tuple
I = < RI, PI, EXTI, SI, LI, LVI >.
RI is the domain of discourse or universe, i.e., a nonempty set that contains the denotations of URI
references and literals in V.
PI is a subset of RI, the properties of I.
EXTI is used to give meaning to properties, and is
a mapping from PI to P(RI × RI).
SI is a mapping from URI references in V to
their denotations in RI.
LI is a mapping from typed literals in V to
their denotations in RI.
LVI is a subset of RI
that contains at least
the set of Unicode strings,
the set of pairs of Unicode strings and language tags,
and the value spaces for each datatype in D.The set of classes CI is defined as
CI =
{ x ∈RI | <x,SI(rdfs:Class)> ∈ EXTI(SI(rdf:type)) },
and the mapping
CEXTI from CI to P(RI) is
defined as
CEXTI(c) = { x∈RI |
<x,c>∈EXTI(SI(rdf:type)) }.
D-interpretations must meet several other conditions,
as detailed in the RDF semantics.
For example, EXTI(SI(rdfs:subClassOf)) must be a transitive
relation and the class extension of all datatypes must be subsets of LVI.

Definition:
Let D be a datatype map that includes datatypes for
rdf:XMLLiteral,
xsd:integer and
xsd:string.
An OWL interpretation,
I = < RI, PI, EXTI, SI, LI, LVI >,
of a vocabulary V,
where V includes the RDF and RDFS vocabularies and the
OWL vocabulary,
is a D-interpretation of V
that satisfies all the constraints in this section.

Note:Elements of the OWL vocabulary that
construct descriptions
in the abstract syntax are given a
different treatment from elements of the OWL vocabulary that correspond to
(other) semantic relationships. The former have only-if semantic
conditions and comprehension principles; the latter have if-and-only-if
semantic conditions. The only-if semantic conditions for the former
are needed to prevent semantic paradoxes and other problems with the
semantics. The comprehension principles for the former and the
if-and-only-if
semantic conditions for the latter are needed so that useful
entailments are valid.

Conditions concerning the parts of the OWL universe and syntactic categories

If E is

then

Note

SI(E)∈

CEXTI(SI(E))=

and

owl:Class

CI

IOC

IOC⊆CI

This defines IOC as the set of OWL classes.

rdfs:Datatype

IDC

IDC⊆CI

This defines IDC as the set of OWL datatypes.

owl:Restriction

CI

IOR

IOR⊆IOC

This defines IOR as the set of OWL restrictions.

owl:Thing

IOC

IOT

IOT⊆RIand IOT ≠ ∅

This defines IOT as the set of OWL individuals.

owl:Nothing

IOC

{}

rdfs:Literal

IDC

LVI

LVI⊆RI

owl:ObjectProperty

CI

IOOP

IOOP⊆PI

This defines IOOP as the set of OWL individual-valued properties.

owl:DatatypeProperty

CI

IODP

IODP⊆PI

This defines IODP as the set of OWL datatype properties.

owl:AnnotationProperty

CI

IOAP

IOAP⊆PI

This defines IOAP as the set of OWL annotation properties.

owl:OntologyProperty

CI

IOXP

IOXP⊆PI

This defines IOXP as the set of OWL ontology properties.

owl:Ontology

CI

IX

This defines IX as the set of OWL ontologies.

owl:AllDifferent

CI

IAD

rdf:List

IL

IL⊆RI

This defines IL as the set of OWL lists.

rdf:nil

IL

"l"^^d

CEXTI(SI(d))

SI("l"^^d) ∈ LVI

Typed literals are well-behaved in OWL.

OWL built-in syntactic classes and properties

I(owl:FunctionalProperty),
I(owl:InverseFunctionalProperty),
I(owl:SymmetricProperty),
I(owl:TransitiveProperty),
I(owl:DeprecatedClass), and
I(owl:DeprecatedProperty)
are in CI.

I(owl:versionInfo),
I(rdfs:label),
I(rdfs:comment),
I(rdfs:seeAlso),
and
I(rdfs:isDefinedBy)
are all in IOAP.
I(owl:imports),
I(owl:priorVersion),
I(owl:backwardCompatibleWith),
and
I(owl:incompatibleWith),
are all in IOXP.

Characteristics of OWL classes, datatypes, and properties

If E is

then if e∈CEXTI(SI(E)) then

Note

owl:Class

CEXTI(e)⊆IOT

Instances of OWL classes are OWL individuals.

rdfs:Datatype

CEXTI(e)⊆LVI

owl:DataRange

CEXTI(e)⊆LVI

OWL dataranges are special kinds of datatypes.

owl:ObjectProperty

EXTI(e)⊆IOT×IOT

Values for individual-valued properties are OWL individuals.

owl:DatatypeProperty

EXTI(e)⊆IOT×LVI

Values for datatype properties are literal values.

owl:AnnotationProperty

EXTI(e)⊆IOT×(IOT∪LVI)

Values for annotation properties are less unconstrained.

owl:OntologyProperty

EXTI(e)⊆IX×IX

Ontology properties relate ontologies to other ontologies.

If E is

then c∈CEXTI(SI(E))
iff c∈IOOP∪IODP and

Note

owl:FunctionalProperty

<x,y1>, <x,y2>
∈ EXTI(c)
implies y1 = y2

Both individual-valued and datatype properties can be functional properties.

If E is

then c∈CEXTI(SI(E))
iff c∈IOOP and

Note

owl:InverseFunctionalProperty

<x1,y>, <x2,y>∈EXTI(c)
implies x1 = x2

Only individual-valued properties can be inverse functional properties.

We will say that l1 is a sequence of
y1,…,yn over C iff
n=0 and l1=SI(rdf:nil)
or n>0 and l1∈IL and
∃ l2, …, ln ∈ IL such that
<l1,y1>∈EXTI(SI(rdf:first)), y1∈C,
<l1,l2>∈EXTI(SI(rdf:rest)), …,
<ln,yn>∈EXTI(SI(rdf:first)), yn∈C, and
<ln,SI(rdf:nil)>∈EXTI(SI(rdf:rest)).

If E is

then <x,y>∈EXTI(SI(E))
iff

owl:complementOf

x,y∈ IOC and CEXTI(x)=IOT-CEXTI(y)

owl:unionOf

x∈IOC and y is a sequence of
y1,…yn over IOC and
CEXTI(x) =
CEXTI(y1)∪…∪CEXTI(yn)

owl:intersectionOf

x∈IOC and y is a sequence of
y1,…yn over IOC and
CEXTI(x) = CEXTI(y1)∩…∩CEXTI(yn)

owl:oneOf

x∈CI and y is a sequence of
y1,…yn over IOT or over LVI and
CEXTI(x) = {y1,..., yn}

Further conditions on owl:oneOf

If E is

and

then if <x,l>∈EXTI(SI(E)) then

owl:oneOf

l is a sequence of y1,…yn over LVI

x∈IDC

owl:oneOf

l is a sequence of y1,…yn over IOT

x∈IOC

Conditions on OWL restrictions

If

then x∈IOR, y∈IOC∪IDC, p∈IOOP∪IODP, and CEXTI(x) =

<x,y>∈EXTI(SI(owl:allValuesFrom))) ∧
<x,p>∈EXTI(SI(owl:onProperty)))

{u∈IOT | <u,v>∈EXTI(p) implies v∈CEXTI(y) }

<x,y>∈EXTI(SI(owl:someValuesFrom))) ∧
<x,p>∈EXTI(SI(owl:onProperty)))

{u∈IOT | ∃ <u,v>∈EXTI(p)
such that v∈CEXTI(y) }

If

then x∈IOR, y∈IOT∪LVI, p∈IOOP∪IODP, and CEXTI(x) =

<x,y>∈EXTI(SI(owl:hasValue))) ∧
<x,p>∈EXTI(SI(owl:onProperty)))

{u∈IOT | <u, y>∈EXTI(p) }

If

then x∈IOR, y∈LVI, y is a non-negative integer, p∈IOOP∪IODP, and CEXTI(x) =

<x,y>∈EXTI(SI(owl:minCardinality))) ∧
<x,p>∈EXTI(SI(owl:onProperty)))

{u∈IOT | card({v ∈ IOT ∪ LV : <u,v>∈EXTI(p)}) ≥ y }

<x,y>∈EXTI(SI(owl:maxCardinality))) ∧
<x,p>∈EXTI(SI(owl:onProperty)))

{u∈IOT | card({v ∈ IOT ∪ LV : <u,v>∈EXTI(p)}) ≤ y }

<x,y>∈EXTI(SI(owl:cardinality))) ∧
<x,p>∈EXTI(SI(owl:onProperty)))

{u∈IOT | card({v ∈ IOT ∪ LV : <u,v>∈EXTI(p)}) = y }

Comprehension conditions
(principles)

The first two comprehension conditions require the existence of the finite
sequences that are used in some OWL constructs.
The third comprehension condition requires the existence of instances of owl:AllDifferent.
The remaining comprehension conditions
require the existence of the appropriate OWL descriptions and data ranges.

5.3. OWL Full

OWL Full augments the common conditions with conditions that force
the parts of the OWL universe to be the same as their analogues in
RDF.
These new conditions strongly interact with the common conditions.
For example, because in OWL Full IOT is the entire RDF domain of discourse,
the second
comprehension condition for lists generates lists of any kind, including
lists of lists.

Definition:
An OWL Full interpretation of a vocabulary V
is an OWL interpretation
that satisfies the following conditions (recall that an OWL interpretation
is with respect to a datatype map).

IOT = RI

IOOP = PI

IOC = CI

Definition:
Let K be a collection of RDF graphs.
K is imports closediff for every triple in any element of K of the form
x owl:imports u .
then K contains a graph that is the result of the RDF processing of
the RDF/XML document, if any, accessible at u into an RDF graph.
The imports closure of a collection of RDF graphs is the smallest
import-closed collection of RDF graphs containing the graphs.

Definitions:
Let K and Q be collections of RDF graphs and D be a datatype map.
Then K OWL Full entails Q with respect to D
iff every OWL Full interpretation with respect to D (of any vocabulary V that includes
the RDF and RDFS vocabularies and the
OWL vocabulary)
that satisfies all the RDF graphs in K
also satisfies all the RDF graphs in Q.
K is OWL Full consistentiff there is
some OWL Full interpretation that satisfies all the RDF graphs in K.

5.4. OWL DL

OWL DL augments the conditions
of Section 5.2
with a separation of the
domain of discourse into several disjoint parts.
This separation has two consequences.
First, the OWL portion of the domain of discourse becomes standard
first-order, in that predicates (classes and properties) and individuals
are disjoint.
Second, the OWL portion of a OWL DL interpretation can be viewed as a
Description Logic interpretation for a particular expressive
Description Logic.

Definition:
A OWL DL interpretation of a vocabulary V
is an OWL interpretation
that satisfies the following conditions (recall that an OWL interpretation
is with respect to a datatype map).

Definitions:
Let K and Q be collections of RDF graphs and D be a datatype map.
Then K OWL DL entails Q with respect to D
iff every OWL DL interpretation with respect to D (of any vocabulary V that includes
the RDF and RDFS vocabularies and the
OWL vocabulary)
that satisfies all the RDF graphs in K
also satisfies all the RDF graphs in Q.
K is OWL DL consistentiff there is
some OWL DL interpretation that satisfies all the RDF graphs in K.

There is a strong correspondence between the Direct Model-Theoretic Semantics and the OWL DL
semantics (but in case of any conflict, the Direct Model-Theoretic
Semantics takes precedence—see the Note at the beginning of Section 5). Basically, an ontology that could be
written in the abstract syntax OWL DL entails another exactly when it
entails the other in the direct semantics. There are a number of
complications to this basic story having to do with splitting up the
vocabulary so that, for example, concepts, properties, and individuals
do not interfere, and arranging that imports works the same.

For the correspondence to be valid there has to be some connection between
an ontology in the abstract syntax with a particular name and the document
available on the Web at that URI. This connection is outside the semantics
here, and so must be specially arranged. This connection is also only an
idealization of the Web, as it ignores temporal and transport aspects of
the Web.

Definition:
Let T be the mapping from the abstract syntax to RDF
graphs from Section 4.1.
Let O be a collection of OWL DL ontologies and axioms and facts in abstract syntax form.
O is said to be imports
closediff for any URI, u, in an imports directive in any ontology in O the
RDF parsing of the document accessible on the Web at u results in
T(K), where K is the ontology in O with name u.

Theorem 1:
Let O and O' be collections of OWL DL ontologies and axioms and facts in abstract syntax form
that are imports closed,
such that their union has a
separated vocabulary
(Section 4.2).
Given a datatype map D that maps xsd:string and
xsd:integer to the appropriate XML Schema
datatypes and that includes the RDF mapping for rdf:XMLLiteral, then
O entails O' with respect to D
if and only if
the translation (Section 4.1) of O
OWL DL entails
the translation of O' with respect to D.
The proof is contained in Appendix A.1.

A simple corollary of this is that, given a datatype map D that maps xsd:string and
xsd:integer to the appropriate XML Schema
datatypes and that includes the RDF mapping for rdf:XMLLiteral,
O is consistent with respect to D
if and only if
the translation of O
is consistent with respect to D.

There is also a correspondence between OWL DL entailment and OWL Full entailment.

Theorem 2:
Let O and O' be collections of OWL DL ontologies and axioms and facts in abstract syntax form
that are imports closed,
such that their union has a
separated vocabulary
(Section 4.2).
Given a datatype map D that maps xsd:string and
xsd:integer to the appropriate XML Schema
datatypes and that includes the RDF mapping for rdf:XMLLiteral, then the translation of O
OWL Full entails
the translation of O' with respect to D
if
the translation of O
OWL DL entails
the translation of O' with respect to D.
A sketch of the proof is contained in Appendix A.2.